The bifurcation set as a topological invariant for one-dimensional dynamics

Fuhrmann, Gabriel; Gröger, Maik; Passeggi, Alejandro

doi:10.1088/1361-6544/abb78c

The bifurcation set as a topological invariant for one-dimensional dynamics

Fuhrmann, Gabriel; Gröger, Maik; Passeggi, Alejandro

Authors

Dr Gabriel Fuhrmann gabriel.fuhrmann@durham.ac.uk
Assistant Professor

Maik Gröger

Alejandro Passeggi

Abstract

For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of (some of) their endpoints. By assuming a global perspective and focusing on the geometric and topological properties of this collection rather than the surviving sets of individual holes, we obtain a novel topological invariant for one-dimensional dynamics. We provide a detailed description of this invariant in the realm of transitive maps and observe that it carries fundamental dynamical information. In particular, for transitive non-minimal piecewise monotone maps, the bifurcation set encodes the topological entropy and strongly depends on the behavior of the critical points.

Citation

Fuhrmann, G., Gröger, M., & Passeggi, A. (2021). The bifurcation set as a topological invariant for one-dimensional dynamics. Nonlinearity, 34(3), Article 1366. https://doi.org/10.1088/1361-6544/abb78c

Journal Article Type	Article
Acceptance Date	Sep 11, 2020
Online Publication Date	Feb 18, 2021
Publication Date	2021-02
Deposit Date	Aug 28, 2021
Publicly Available Date	Sep 8, 2021
Journal	Nonlinearity
Print ISSN	0951-7715
Electronic ISSN	1361-6544
Publisher	IOP Publishing
Peer Reviewed	Peer Reviewed
Volume	34
Issue	3
Article Number	1366
DOI	https://doi.org/10.1088/1361-6544/abb78c

Files

Published Journal Article (840 Kb)
PDF

Publisher Licence URL
http://creativecommons.org/licenses/by/3.0/

Copyright Statement
Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.